Why does the Pope include himself among the sinners?

What the pope is saying is correct. A sin is whenever you do anything wrong at all, period. In Romans 3:23-24 (ESV), it is written,

"...for all have sinned and fall short of the glory of God, and are justified by his
grace as a gift, through the redemption that is in Christ Jesus..."

Christianity is all about the fact that all of us have sinned, so all of us must use God's mercy and grace through Christ Jesus's death and resurrection to obtain eternal life. Also read 1 John 1:8-9 and Romans 3:9-10.

He is calling himself a sinner because his doctrine determines him to be so.

Many Christians will also call themselves sinners. Likewise, other believers know themselves to be saints. What's important though is what the bible says about them that believe.

In most of Paul's letters he calls the people in the church (the believers) "saints". He even addresses the Corinthian church as a people who are called to be saints. Remember, this was one messed up church - there were men who were having sexual relations with their father's wife among other perversions. Yet their calling was to "be" saints.

Think about that. Do you think that men who had had this horrid stuff going on in their lives qualify to be saints? According to scripture - YES.
The very same that Jesus called to salvation.

Unto the church of God which is at Corinth, to them that are sanctified in Christ Jesus, called to be saints, with all that in every place call upon the name of Jesus Christ our Lord, both theirs and ours:

Here's another verse that reveals saints as not perfect people but people who are being perfected. That's a huge difference. Saints can sin, but it's not right behaviour.

For the perfecting of the saints, for the work of the ministry, for the edifying of the body of Christ: - Eph 4:12

The word saint means in greek - most holy thing. Counter intuitively the word Holy also means the same thing, Saint or Holy. So, what does it really mean to be Holy or to be a Saint?

Most Christians think that to be Holy means to to good, to keep God's laws. But:

All have sinned and fallen short of the glory of God - Romans 3:23

So this is not an accurate definition of what Holiness is. And likewise, I haven't met a christian yet who hasn't sinned after becoming a believer.

Whatever it means to be Holy or a Saint must have nothing to do with us, because no one qualifies on their own goodness. Romans makes that pretty clear in the first couple of chapters. We have nothing to boast about and the only way to be saved is through faith in God's saviour Jesus.

You'll also notice that in the OT, certain things in the temple were Holy.
Q. What made them Holy? My definition of Holy is set apart. Likewise, believers (christians) are set apart. We are told to be in the world and not of the world. We are told to be ambassadors for Christ 2 Cor 5:20. We now represent Him. To do that, we need to know who were are in Christ.

Here's a glimpse which is a little too much to write in a few paragraphs but: When we believe we are born again. We receive a brand knew spirit in our inner man (the new man) and we now have the mind of Christ. 1 Cor 2:16. The old man is dead (your old spirit) The Spirit and the Soul are not the same thing. Your spirit is the new creation. Your soul and body are yet to be renewed, but your spirit is perfect, it is holy.

Sin is not confined to a mere ten commandments. Jesus Himself expanded on this in the Sermon on the Mount. There He asserted that it wasn't just murder that was sinful, but hatefulness in the heart. It wasn't just adultery that was immoral, but lust in the heart.

Indeed, we are all sinners—and really bad sinners for that matter. Romans 1-3 asserts this pretty clearly.

The Pope actually has someone to which he confesses his own sins according to this question.

To be sure, the Pope likely does not engage in anything considered too bad. It should be noted, though, that the sin of covetousness is, in fact, one of the ten commandments. So, an inappropriate desire to possess anything that is not rightfully yours and belongs to another, is a sin. 

The doctrine of original sin says that every human being is a sinner.

Indeed, the Bible also says:

Romans 3:10

As it is written: "There is no one righteous, not even one;

and

Isaiah 56:3

All we like sheep have gone astray; we have turned—every one—to his own way; and the LORD has laid on him the iniquity of us all.

and Isaiah 64:6

We have all become like one who is unclean, and all our righteous deeds are like a polluted garment. We all fade like a leaf, and our iniquities, like the wind, take us away.

Indeed, only Jesus escaped this sin - as Hebrews 4:15, in describing the uniqueness of Jesus writes:

For we do not have a high priest who is unable to sympathize with our weaknesses, but one who in every respect has been tempted as we are, yet without sin.

Here, the obvious implication is that even High Priests (and their modern counterparts) sin.

Indeed, even the canons of the church explicitly clarify this point:

Trent-Justification Canon 1. "If anyone says that man can be justified before God by his own works, whether done by his own natural powers or through the teaching of the law, without divine grace through Jesus Christ, let him be anathema."

With the possible exception of Pope Joan^*, every Pope is man. As man, he is subject to the same consideration.

The idea of total depravity and the universality of sin says that everyone (and that includes a Pope who knows his scripture well!) is both a sinner and under judgment. Indeed, this idea gives birth to many popular sayings on the matter, such as:

• A saint is just a sinner who falls down, and gets back up

• Evangelism is just one beggar telling another where to find bread - D.T. Niles.

While it is tempting to chalk this up to humility the truth of the matter is that any priest worth his salt already knows this in his soul.

Oh, and finally, before you ask, yes, Pope Francis has his own confessor.

^{*The Pope Joan thing was a joke, in case you are wondering.}